algebra i & mathematics i real numbers (). these units were created by mathematics educators...

Algebra I & Mathematics IAlgebra I & Mathematics I

Student EditionStudent Edition

Real Numbers

Real NumbersPathways to

Common CoreMathematics

Algebra I & Mathematics I

Student Edition

S T E MScience Technology Engineering Mathematics

Real Numbers- Student Edition

ISBN 978-1-940836-05-8

© 2014

For Information Contact:Center for Mathematics, Science, and Technology210 W. Mulberry StIllinois State UniversityNormal, IL 61790309-438-3089http://cemast.illinoisstate.edu

All rights reserved. This book or parts thereof may not be reproduced or used in any form or by any means-graphic, electronic, or mechanical, including photocopying, recording, taping, or information storage and retrieval systems-without written permission of the publisher. Making copies of this book, or any portion, is a violation of United States copyright laws.

Printed in the United States of America

Table of Contents

ACKNOWLEDGEMENTS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . v

PREFACE . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . vi

CHAPTER 1 | Relationships Between Quantities . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

Come Fly with Me . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

The Grass is Always Greener . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

Functional Functions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19

Par for the Math Course . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25

Tips . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37

CHAPTER 2 | Reasoning With Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45

Farming Revenue . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47

Future of the Landfill . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53

Life is a Highway . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61

Open Campus? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71

Party Time . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75

CHAPTER 3 | Linear And Exponential Relationships . . . . . . . . . . . . . . . . . . . . . . . . . 81

Affordable on Any Budget . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83

Fuel Valve Metering Ports . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91

Eating the Trees . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97

Spaghetti Bridge . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103

Teen Driving Premiums . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111

CHAPTER 4 | Descriptive Statistics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117

Teen Policy Costs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119

Parking Lot Math . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 127

How Long Will You Wait? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 135

Under New Management . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 141

Gender Health . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 151

Real Numbers: Pathways to Common Core Mathematics iii

CHAPTER 5 | Expressions And Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 157

Dino-mite . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 159

Medication Math . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 165

Bicycle Math . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 171

Ballroom Math . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 177

CHAPTER 6 | Quadratic Functions And Modeling . . . . . . . . . . . . . . . . . . . . . . . . . . 183

Latitude Adjustment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 185

Kick the Football . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 191

Plastic Money . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 197

Waste Not . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 207

Students vs. Zombies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 213

CHAPTER 7 | Congruence, Proof, & Constructions . . . . . . . . . . . . . . . . . . . . . . . . . . 219

Let’s Ship Something Together . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 221

Fly Me Away . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 233

How Aligned is Your Spine? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 243

Custom Hot Rod Shop . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 249

Taxi! Taxi! . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 259

CHAPTER 8 | Connecting Algebra & Geometry . . . . . . . . . . . . . . . . . . . . . . . . . . . 265

CAD/CAM Project . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 267

Are You Square? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 275

Piles of Tiles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 281

Elbow Room . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 287

iv Real Numbers: Pathways to Common Core Mathematics

Maria Adams

Chelsea Baird

Lindsay Battle

Pete Beke

Nik Butenhoff

Carrie Daly

David Daraskevich

Leann Dunlap

Janet Erb

Jeff Fisher

Cecile Flowers

Danni Gallagly

Nick Gaona

Judy Hutchinson

Amber Misek

Rick Morton

Lauren Nelson

Mark Ornatek

Diane Powell

Jennifer Sales

Jai Sharma

Ashley Spindler

Marissa Williams

Abby Newcomb

Welcome to Real Numbers: Pathways to Common Core Mathematics, a curriculum that provides unique and engaging mathematics learning experiences in Algebra and Geometry. Using mathematical modeling, students will learn math concepts through authentic, real-world applications.

At the beginning of each unit in this book, you will see a “big-picture” context for those applications anchored in U.S. Department of Education Career Clusters and Pathways (www.careerclusters.org). Each unit is also aligned with Common Core State Standards for Mathematics (www.corestandards.org). These units were created by mathematics educators based on visits to businesses in the community during which they explored ways in which mathematics is used in the real world.

The eight chapters of Real Numbers comprise the entirety of the content recommended by the Common Core State Standards for Mathematics for both the Algebra I and the Mathematics I courses. Chapters 1 through 4 are common

to both courses. Chapters 5 and 6 are recommended for the Algebra I course only. Chapters 7 and 8 are recommended for the Mathematics I course only.

It should be noted that Real Numbers does not simply use the context provided by the Career Custer as the basis of a series of story problems. Instead, this curriculum teaches the content through its application. For example, students learn how to calculate ratios by being given the problem of mixing fertilizer for a golf course. This educational process is made possible through the use of Learning Cycles.

About the Learning CycleEvery lesson in this book is organized into a learning cycle. A learning cycle consists of four parts; Try It, Discuss It, Apply It, and Expand It. In some cases, there may be multiple Try It sections, each followed by a Discuss It section. There is always one Apply It and Expand It section in each learning cycle.

Section Description Student Role Teacher Role

Try ItRemember and begin to

understand

Students engage the content through an activity. They are required to build, gather data, look for patterns, etc. Students are prompted to remember previous learning and connect it to new concepts as their understanding grows.

Work in groups or individually, follow the written directions to accomplish the task, collect and record data, make predictions

Facilitate the learning activity. Provide necessary materials and tools. Ask questions, but do not answer them. Assure safe practices.

Discuss ItUnderstand and begin to apply

A series of questions designed to clarify concepts and identify misconceptions. Assesses understanding and probes applications.

Complete the questions as assigned. This could be orally as a large group, written individually, or written as a group. Students compare data, form generalizations, write in journals.

Monitor student responses and ask clarifying questions. Assess understanding. Corrects misconceptions.

Apply ItApply and begin to analyze

Usually a continuation of an activity that requires the application of the concepts learned in the previous two sections. Requires analysis and evaluation of concepts.

Complete the assigned task, utilizing knowledge gained from the previous sections of the learning cycle.

Facilitate the activity. Provide necessary tools and materials. Monitor progress. Inject leading questions as appropriate. Assure safe practices. Corrects lingering misconceptions.

Expand ItAnalyze and begin to evaluate

and create

A few ideas for taking the concept further, possibly a different, but related, application. These tasks require analysis, evaluation, and creation of new applications and connections.

Use the ideas in this section as a foundation for further research.

These ideas provide opportunity for further study. Allow students to explore these concepts on their own. Many will be so interested in the topic that independent learning will occur without prompting.

Welcome!

Preface

vi Real Numbers: Pathways to Common Core Mathematics

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AssessmentThere are several ways to view assessment of student learning in mathematics. Many school districts have a very specific and rigid system of assessment utilizing regularly scheduled standardized tests consistent with state assessment strategies. Other districts are a bit more flexible utilizing teacher-written tests throughout the year culminating with state-mandated standardized testing for certain grades. Regardless of the local process, the lessons in the Real Numbers: Pathways to Common Core Mathematics can be quite useful.

1 All assessments are designed to indicate the level of understanding of a particular topic. When it is clear that a particular topic is not mastered, it makes little sense to teach it again using the same strategies. The lessons in this book provide an alternative. Many districts require that students pass Algebra I for graduation. Well over 70% of the students dropping out of high school claim it is because they cannot pass Algebra I. The lessons in this book address Algebra I content by utilizing student experiences and interests to help them realize the importance of mathematics and lead to passing the algebra graduation requirement.

2 The lessons in this book can be used as a pre-test. Perhaps when first addressing a topic, students are given a lesson from this text. As they progress through the learning cycle, misunderstandings become apparent. The teacher then knows what to cover in more depth. This could occur during the learning cycle or after it has been completed.

3 The lessons in this book can be used as a post-test. After receiving instruction on a particular topic, the corresponding learning cycle could be assigned. Successful completion of the activities could be used to determine mastery.

4 The teacher may decide to write assessments consistent with local policies that measure important objectives aligned with the philosophy of the learning cycle.

5 The teacher may decide to use the learning cycle as a means to teach important objectives that will be assessed using district or state standardized assessment tools.

6 The activities from the Apply It or the Expand It sections of the learning cycle could be developed into an assessment. A rubric could be created to describe and quantify the quality of work.

Importance of Real-World ConnectionsEngaging students in experiences that are connected to the real world is critically important to learning. In her publication, “How Children Learn,” Stella Vosniadou discusses research that identifies key factors in effective teaching and learning, including:

• Learning becomes more meaningful when the lessons are applied to real-life situations.

• People learn best when they participate in activities that are perceived to be useful in real life and are culturally relevant.

• Learning requires the active, constructive involvement of the learner. 1

Dr. Bill Daggett has provided strong leadership for relevance in education. His Rigor and Relevance Framework offers a compelling visual argument for authentic connections and rigorous curriculum.1 Vosniadou, Stella. How Children Learn. International Academy of Education. 2001. <http://tinyurl.com/6m82jvk>.

Students gather and share bitsof knowledge and information.Students are primarily expectedto remember or understand thisknowledge.

Students use acquiredknowledge to solve problems,design solutions, andcomplete work. The highestlevel of application is toapply knowledge to new andunpredictable situations.

Students extend and refinetheir acquired knowledge tobe able to use that knowledgeautomatically and routinely toanalyze and solve problemsand create solutions.

Students have the competenceto think in complex ways andto apply their knowledge andskills. Even when confrontedwith perplexing unknowns,students are able to useextensive knowledge and skillto create solutions and takeaction that further developstheir skills and knowledge.

Application Model

International Center for Leadership in Education, http://www.leadered.com/rrr.html

Preface

Real Numbers: Pathways to Common Core Mathematics vii

PATHWAYS TO COLLEGE & CAREER READINESS

TM

One of the keys to improving student achievement is providing students with relevant contexts for studying and learning. Career Clusters™ do exactly this by linking school-based learning with the knowledge and skills required for success in the workplace. The National Career Clusters™ Framework is comprised of 16 Career Clusters™ and related Career Pathways to help students of all ages explore different career options and better prepare for college and career.

Each Career Cluster™ represents a distinct grouping of occupations and industries based on the knowledge and skills they require. The 16 Career Clusters™ and related Career Pathways provide an important organizing tool for schools to develop more effective programs of study (POS) and curriculum.

Quoted from: http://www.careertech.org/career-clusters/glance/careerclusters.html

Preface

viii Real Numbers: Pathways to Common Core Mathematics

Learning Cycle Content Context

Chapter 1 Algebra I and Mathematics I

Chapter 1Relationships Between QuantitiesCome Fly With Me

Objectives• Observe,collect,andrecorddatainnumericaland

graphical forms.

• Useconversionstosolveproblems.

Common Core StandardsQUANTITIESReason quantitatively and use units to solve problems.N-Q-1, 2, 3

Rockford International AirportRockford, ILTransportation

Chapter 1 Relationships Between QuantitiesThe Grass Is Always Greener

Objectives• Analyzeproportionalrelationshipsandpercentages,

and use them to solve real-world and mathematical problems

• Useunits,conversions,andunitratestosolveproblems and make decisions

• Determinetheeffectsofchangingpartsofawhole

Common Core StandardsSEEINGSTRUCTUREINEXPRESSIONSInterpret the structure of expressions. A-SSE 1a

QUANTITIES Reason quantitatively and use units to solve problems. N-Q 1, 2

Elliot Golf Course, Rockford, ILAgriculture, Food, and Natural Resources

Chapter 1 Relationships Between QuantitiesFunctional Functions

Objectives• Understandthatafunctionisarulethatassignsto

each input exactly one output

Common Core StandardsINTERPRETINGFUNCTIONSUnderstand the concept of a function and use function notationF-IF 1

Woodward, Rockford, ILManufacturing

Chapter 1 Relationships Between QuantitiesPar for the Math Course

Objectives• Usecommonareaformulastoapproximatethearea

of irregular shapes

Common Core StandardsQUANTITIESReason quantitatively and use units to solve problemsN-Q 1, 2, 3

GEOMETRYApply geometric concepts in modeling situationsG-MG 1

Elliot Golf Course, Rockford, ILArchitecture and Construction

Content and Context of Each Learning Cycle

Preface

Real Numbers: Pathways to Common Core Mathematics ix



Chapter 1Relationships Between QuantitiesTips

Objectives• Writeexpressionsinequivalentformstosolve

problems

Common Core StandardsSEEINGSTRUCTUREINEXPRESSIONSInterpret the structure of expressions. A-SSE 1

Giovanni’s Restaurant, Rockford, ILAgricultural, Food, and Natural Resources


Chapter 2Reasoning with Equations and InequalitiesFarming Revenue

Objectives• Writeequationsfromreal-lifesituations.• Understandandexplainverballyhowtosolvean

equation.• Solveequations

Common Core StandardsCREATINGEQUATIONSCreate equations that describe numbers or relationships.A-CED2

Boone County Farm Bureau/ Central Grain Company, Belvidere, ILAgriculture, Food, and Natural Resources

Chapter 2Reasoning with Equations and InequalitiesFuture of the Landfill

Objectives• Understandhowtocalculatevolume• Useratiostopredictquantities• Collectandorganizedata

Common Core StandardsREASONINGWITHEQUATIONSANDINEQUALITIESUnderstand solving equations as a process of reasoning and explain the reasoningA-REI 1Solve equations and inequalities in one variableA-REI 3QUANTITIESReason quantitatively and use unites to solve problemsN-Q 1, 2

Rock River Environmental Services, Rockford, ILArchitecture and Construction

Chapter 2Reasoning with Equations and InequalitiesLife is a Highway

Objectives• Writeandsolveinequalities

Common Core StandardsSTATISTICSANDPROBABILITYMake inferences and justify conclusions from sample surveys, experiments, and observational studiesS-IC 6 CREATINGEQUATIONSCreate a number of equations that describe numbers or relationshipsA-CED1REASONINGWITHEQUATIONSANDINEQUALITIESSolve equations and inequalities in one variableA-REI 3

US Bank, Rockford, ILFinance

Preface

x Real Numbers: Pathways to Common Core Mathematics



Chapter 2Reasoning with Equations and InequalitiesOpen Campus?

Objectives• Analyzeasituationanddetermineconstraintsthat

affect the situation.• Setupandsolveasystemoflinearinequalities.• Identifyasetofsolutionsfromafeasibleregion.

Common Core StandardsINTERPRETINGFUNCTIONSInterpret functions that arise in applications in terms of the context.F-IF 4, 7 BUILDINGFUNCTIONSBuild a function that models a relationship between two quantities.F-BF 1LINEAR,QUADRATIC,ANDEXPONENTIALMODELSInterpret expressions for functions in terms of the situation they model.F-LE 5REASONINGWITHEQUATIONSANDINEQUALITIESRepresent and solve equations and inequalities graphically.A-REI 10, 12

BelvidereNorthHighSchoolArchitecture

Chapter 2Reasoning with Equations and InequalitiesParty Time

Objectives• Determinearelationship.• Solveonestepequationsresultingfromrealworld

situations. • Determineindependentanddependentvariables

Common Core StandardsSEEINGSTRUCTUREINEXPRESSIONSInterpret the structure of expressions.A-SSE 1 CREATINGEQUATIONSCreate equations that describe numbers or relationships.A-CED1INTERPRETINGFUNCTIONSAnalyze functions using different representationsF-IF 7

Giovanni’s Restaurant, Rockford, ILFinance


Chapter 3Linear and Exponential RelationshipsAffordable on Any Budget

Objectives• Interpretfunctionsthatariseinapplicationsinterms

of context• Constructandcomparelinear,quadraticand

exponential models and solve problems• Interpretexpressionsforfunctionsintermsofthe

situation they model

Common Core StandardsSEEINGSTRUCTUREINEXPRESSIONSInterpret the structure of expressionsA-SSE 1b

US Bank,Rockford, ILFinance

Preface

Real Numbers: Pathways to Common Core Mathematics xi



Chapter 3Linear and Exponential RelationshipsFuel Valve Metering Ports


graphical forms.• Comparelinear,quadratic,andexponentialmodels

of fuel valve openings.

Common Core StandardsFUNCTIONSConstruct and compare linear, quadratic, and exponential models and solve problemsF-LE 1a-c, 2, 3, 4

Woodward, Loves Park, ILManufacturing

Chapter 3Linear and Exponential RelationshipsEating The Trees


graphical forms.• Usetheprincipleofhalf-lifetodeterminehowlong

it will take for the emerald ash borer to eradicate all the ash trees in the Midwest.

Common Core StandardsLINEAR,QUADRATIC,ANDEXPONENTIALMODELSInterpret expressions for functions in terms of the situation they model.F-LE 5

Elliot Golf Course,Rockford, ILAgriculture, Food and Natural Resources

Chapter 3Linear and Exponential RelationshipsSpaghetti Bridge

Objectives• Writeanequationofabestfitlineofasetofdata• Usethatequationtoanalyzethesituation

Common Core StandardsCREATINGEQUATIONSCreate equations that describe numbers or relationshipsA-CED2Represent and solve equations and inequalities graphicallyA-REI 10LINEAR,QUADRATIC,ANDEXPONENTIALMODELSInterpret functions that arise in applications in terms of the contextF-IF 4, 5Build a function that models a relationship between two quantitiesF-BF 1Interpret expressions for functions in terms of the situation they modelF-LE 5

McClure Engineering,Rockford, ILArchitecture and Construction

Chapter 3Linear and Exponential RelationshipsTeen Driving Premiums

Objectives• Understandthepropertiesoffunctions:f(x)±g(x)=(f±g)(x)• Solveproblemsusingtheadditionandsubtractionof functions

Common Core StandardsINTERPRETINGFUNCTIONS

AW Anderson Agency Loves Park, ILFinance

…continued on next page

Preface

xii Real Numbers: Pathways to Common Core Mathematics


Chapter 3 Algebra I and Mathematics I …continued

Chapter 3Linear and Exponential RelationshipsSpaghetti Bridge

Understand the concept of a function and use function notationF-IF 1, 2Analyze functions using different representationsF-IF 9BUILDINGFUNCTIONSBuild a function that models a relationship between two quantitiesF-BF 1a-c

McClure Engineering,Rockford, ILArchitecture and Construction


Chapter 4DescriptiveStatisticsTeen Policy Costs

Objectives• Understandthepropertiesoffunctions• Addandsubtractfunctions

Common Core StandardsCREATINGEQUATIONSCreate equations that describe numbers or relationshipsA-CED1LINEAR,QUADRATIC,ANDEXPONENTIALMODELSUnderstand the concept of a function and use function notationF- IF 1a-c, 2, 3

AW Anderson Agency Loves Park, ILFinance

Chapter 4DescriptiveStatisticsParking Lot Math

Objectives• Gatherandrecorddata• Understandhowrandomsamplescanbeappliedto

the population• Explorebivariatestatistics• Displaystatisticaldataappropriately

Common Core StandardsINTERPRETINGCATEGORICALANDQUANTITATIVEDATASummarize, represent, and interpret data on two categorical and quantitative variables.S-ID5Understand and evaluate random processes underlying statistical experiments.S-IC 1 Understand independence and conditional probability and use them to interpret data.S-CP 4

BelvidereNorthHighSchoolBelvidere, ILManufacturing

Chapter 4DescriptiveStatisticsHow Long Will You Wait?

Objectives• Computemeasuresofcentraltendencyandvariance• Analyzedataandmakedecisionsbasedonmeasures

of variance

Common Core StandardsINTERPRETINGCATEGORICALANDQUANTITATIVEDATASummarize, represent, and interpret data on a single count or measurement.S-ID2,4

Giovanni’s Restaurant,Rockford, ILCustomerService—Hospitality

Preface

Real Numbers: Pathways to Common Core Mathematics xiii



Chapter 4DescriptiveStatisticsUnder New Management

Objectives• Summarize,represent,andinterpretdataontwo

quantitative variables• Interpretlinearmodelsandusethemodeltomake

predictions• Interpretthemeaningofweightedvariablesinan

equation• Constructalineofbestfit

Common Core StandardsINTERPRETINGCATEGORICAL&QUANTITATIVEDATASummarize, Represent, and interpret data on two categorical and quantitative variables.S-ID6Interpret linear models.S-ID7SEEINGSTRUCTUREINEXPRESSIONSInterpret the structure of expressions. A-SSE 1a

RockfordIceHogsRockford, IL Finance

Chapter 4DescriptiveStatisticsGender Health

Objectives• Createafrequencychartandhistogram• Compareandcontrastdata• Analyzedataandgraphs• Usedatainordertosupportanargument

Common Core StandardsINTERPRETINGCATEGORICALANDQUANTITATIVEDATASummarize, represent, and interpret data on two categorical and quantitative variables.S-ID5Understand and evaluate random processes underlying statistical experiments.S-IC 1 Understand independence and conditional probability and use them to interpret data.S-CP 4

RockfordMemorialHospitalRockford, ILHealthScience

Chapter 5 Algebra I only

Chapter 5Expressions and EquationsDino-Mite

Objectives• Createandsolvesystemsofequations• Understandthesystemsofequationsusinggraphs

and tables

Common Core StandardsREASONINGWITHEQUATIONSANDINEQUALITIESUnderstand solving equations as a process of reasoning and explain the reasoningA-REI 6, 11

Burpee Museum, Rockford, ILResearchandDevelopment,NaturalSciencesPathway

Preface

xiv Real Numbers: Pathways to Common Core Mathematics



Chapter 5Expressions and EquationsMedication Math

Objectives• Identifyindependentanddependentvariablesin

order to graph linear relationships• Createequationsthatdescribenumbersor

relationships

Common Core StandardsQUANTITIESReason quantitatively and use units to solve problemsN-Q 2 CREATINGEQUATIONS Create equations that describe numbers or relationshipsA-CED1INTERPRETINGFUNCTIONSAnalyze functions using different representationsF-IF 7a-e

RockfordMemorialHospitalRockford, ILHealthSciences

Chapter 5Expressions and EquationsBicycle Math

Objectives• Collectinformationandwriteequations.• Rearrangeformulastogiveformulasnewmeanings.

Common Core StandardsCREATINGEQUATIONSCreate equations that describe numbers or relationships.A-CED4

City of Loves ParkLoves Park, ILTransportation

Chapter 5Expressions and EquationsBallroom Math

Objectives• UseAlgebratilestofactorpolynomialsintheform

x² +bx +c.

Common Core StandardsSEEINGSTRUCTUREINEXPRESSIONSWrite expressions in equivalent forms to solve problems. A-SSE-3aPerform arithmetic operations on polynomials.A-APR-1

Giovanni’s Restaurant, Rockford, ILArchitecture/Construction


Chapter 6 Quadratic Functions and ModelingLatitude Adjustment

Objectives• Researchweatherdataandcreateascatterplotof

data• Lookatquadraticregressionlinesandexaminethe

parts of the quadratic equation

Common Core StandardsCREATINGEQUATIONSCreate equations that describe numbers or relationshipsA-CED1

Chicago Rockford International AirportRockford, ILTransportation

Preface

Real Numbers: Pathways to Common Core Mathematics xv



Chapter 6 Quadratic Functions and ModelingKick The Football

Objectives• Interpretfunctionsthatariseinapplicationsinterms

of a context• Analyzefunctionsusingdifferentrepresentations• Buildafunctionthatmodelsarelationshipbetween

two quantities

Common Core StandardsCREATINGEQUATIONSCreate equations that describe numbers or relationshipsA-CED2,3,4INTERPRETINGFUNCTIONSInterpret functions that arise in applications in terms of the contextF-IF 4

BelvidereNorthHighSchoolBelvidere, ILResearchandDevelopment

Chapter 6 Quadratic Functions and ModelingPlastic Money

Objectives• Representasetofdatainatable,graph,and

function form• Identifyanon-linearequationbyanalyzingthedata• Explorethedifferenttypesoffunctions

Common Core StandardsLINEAR,QUADRATIC,ANDEXPONENTIALMODELSConstruct and compare linear, quadratic, and exponential models and solve problemsF-LE 1a-c, 2Interpret expressions for functions in terms of the situation they modelF-LE 5

US Bank, Rockford, ILFinance

Chapter 6 Quadratic Functions and ModelingWaste Not

Objectives• Givendata,determinethedifferencebetweena

linear and exponential function• Givendata,writeanequationforanexponential

function

Common Core StandardsFUNCTIONSBuild a function that models a relationship between two quantities.F-BF 1a-c, 2

Rock River Environmental Services, Rockford, ILArchitecture and Construction

Chapter 6 Quadratic Functions and ModelingStudents vs. Zombies

Objectives• Constructexponentialmodelsusingdifferentbases.• Interpretexpressionsforfunctionsintermsof

situations they model.

Common Core StandardsBUILDINGFUNCTIONSBuild a function that models a relationship between two quantities.F-BF 1a-c, 2

RockfordMemorialHospitalRockford, ILHealthScience

Preface

xvi Real Numbers: Pathways to Common Core Mathematics

Preface


Chapter 7 Mathematics I only

Chapter 7 Congruence, Proof, and ConstructionsLet’s Ship Something Together

Objectives• Experimentwithtransformationsintheplane• Understandcongruenceintermsofrigidmotions

Common Core StandardsCONGRUENCEExperiment with transformations in the planeG-CO5Understand congruence in terms of rigid motionsG-CO6

LowesDistributionCenterRockford, ILTransportation,DistributionandLogistics

Chapter 7Congruence, Proof, and ConstructionsFly Me Away

Objectives• Connectingalgebraandgeometrythrough

coordinates• Usecoordinatestoprovesimplegeometrictheorems

algebraically

Common Core StandardsCONGRUENCEExperiment with transformations in the plane.G-CO5

Rockford International AirportRockford, ILTransportation

Chapter 7Congruence, Proof, and ConstructionsHow Aligned Is Your Spine?

Objectives• Constructparallelandperpendicularlines.

Common Core StandardsCONGRUENCEMake geometric constructions.G-CO12

HulsebusChiropracticRockford, ILHealthScience

Chapter 7Congruence, Proof, and ConstructionsCustom Hot Rod Shop

Objectives• Formadeductiveargument

HopperstadCustoms,Belvidere, ILTransportation

Chapter 7Congruence, Proof, and ConstructionsTaxi! Taxi!

Objectives• Constructingconvincingarguments• Forminghypothesis• Generalizingpatterns

City of Rockford Engineering Department,

Rockford, ILTransportation

Real Numbers: Pathways to Common Core Mathematics xvii

Preface


Chapter 8 Mathematics I only

Chapter 8Connecting Algebra and GeometryCAD/CAM Project

Objectives• Locatepointsonacoordinateplaneandwritethem

as ordered pairs

Common Core StandardsFUNCTIONSDefine,evaluate,andcomparefunctions.8.F 1CONGRUENCEExperiment with transformations in the planeG-CO2

Elite Tool and Wire Rockford, ILManufacturing

Chapter 8Connecting Algebra and GeometryAre You Square?

Objectives• UsethePythagoreanTheoremtomodelrealworld

problems.

Common Core StandardsQUANTITIESReason quantitatively and use units to solve problems.N-Q 1, 3SIMILARITY,RIGHTTRIANGLES,ANDTRIGONOMETRYDefinetrigonometricratiosandsolveproblemsinvolving right triangles.G-SRT 8GEOMETRYUnderstand and apply the Pythagorean Theorem.8.G.7

McClure EngineeringRockford, ILArchitecture&Construction

Chapter 8Connecting Algebra and GeometryPiles of Tiles

Objectives• Usemeasurementstocalculatesurfacearea• Applyconceptofsurfaceareatoreal-life

situation

Common Core StandardsQUANTITIESReason quantitatively and use units to solve problems.N-Q 3

BelvidereNorthHighSchoolBelvidere, ILArchitecture and Construction

Chapter 8Connecting Algebra and GeometryElbow Room

Objectives• Findarclength• Findtheareaofasector• Findtheareaofgeometricshapes

Common Core StandardsQUANTITIESReason quantitatively and use units to solve problems.N-Q 1, 2, 3MODELINGWITHGEOMETRYApply geometric concepts in modeling situationsG-MG 2Find arc lengths and areas of sectors of circles.G-C 5

Giovanni’s Restaurant, Rockford, ILArchitecture and ConstructionAgriculture, Food, and Natural Resources

xviii Real Numbers: Pathways to Common Core Mathematics

Algebra I & Mathematics IAlgebra I & Mathematics I

Teacher EditionTeacher Edition

Real Numbers

ISBN 978-1-940836-06-5 — Teacher Edition

ISBN 978-1-940836-05-8 — Student Edition

© 2014

For Information Contact: Center for Mathematics, Science, and Technology 210 W. Mulberry St Illinois State University Normal, IL 61790 309-438-3089 http://cemast.illinoisstate.edu

S T E M Science Technology Engineering Mathematics

Real Numbers

All rights reserved. This file is intended for use by teachers in Illinois only. Any reproduction, or use in any

form or by any means — graphic, electronic, or mechanical, including photocopying, recording, taping,

information storage and retrieval systems — by anyone other than Illinois teachers without written

permission of the publisher is prohibited.

Printed in the United States of America.

Relationships Between Quantities

2 Come Fly with Me!

COMMON CORE StandardsQUANTITIESReason quantitatively and use units to solve problems.N-Q 1 Use units as a way to understand problems and to guide the solution of multi-step problems; choose and interpret units consistently in formulas; choose and interpret the scale and the origin in graphs and data displays.N-Q 2 Define appropriate quantities for the purpose of descriptive modeling.N-Q 3 Choose a level of accuracy appropriate to limitations on measurement when reporting quantities.

Standards for Math Practice1. Make sense of problems and persevere in solving them. 2. Reason abstractly and quantitatively.3. Construct viable arguments and critique the reasoning of others. 4. Model with mathematics.5. Use appropriate tools strategically.6. Attend to precision. 7. Look for and make use of structures.8. Look for and express regularity in repeated reasoning.

Chapter 1


Come Fly with Me!

Illinois Pathway: Transportation

Objectives

• Observe, collect, and record data in numerical and graphical forms.• Use conversions to solve problems.

Real Numbers: Pathways to Common Core Mathematics 3

Teacher Edition

IntroductionPilots need to know whether they are going to make it to their destination without having to refuel. There are many factors that can determine what type of plane they will fly. Show the following video clip to get the students interested in the topic and the various single-engine planes that are available. http://www.youtube.com/watch?v=B1wj5W-CdqU

4 Come Fly with Me!

T�� I�

1 Research four different types of four-seater single engine airplanes and collect data on the following: speed in knots, maximum range in nautical miles, and usable fuel. Fill in table below. (knot = nautical miles/hr)

2 Calculate the gal/hr of each airplane by using the data you collected on speed, max range and usable fuel. (Use the knowledge that knots are nm/hr, max range is nautical miles and usable fuel is gallons.)

Plane Type Speed (knots) Max Range Usable Fuel Gallons/hr

3 Create a graphical representation below using the plane type and gal/hr.

Chapter 1

One website that is a good source for the needed data is:

http://planes.findthebest.com/d/c/Propeller-Plane

knot = nautical miles/hr

Graphs will vary.

D�� I�

1 When comparing the planes in your bar graph, which plane was the most economical to fly? Least economical?

2 Which plane would you choose to fly? Why or why not?

T�� I�

1 Make a graphical representation to represent the four planes’ gal/hr and speed.

2 Which is the dependent and independent variable? Does it matter?

D�� I�

1 Did you find any relationship between gal/hr and speed? If so, what kind?

2 Do you notice a trend?



Answers will vary.

Answers will vary.

Graphs will vary.

Independent: speed, Dependent: gal/hr

A negative relationship

As speed increases, gal/hr decreases

6 Come Fly with Me!

3 Based on your graph, estimate the speed of an airplane if you only want to use 22 gal/hr.


Answers will vary.

T�� I�

1 Choose one of your airplanes from your table above.

2 Place a point on the y-axis to indicate the airspeed.

3 Zero on the x-axis means no wind. The positive x represents a tailwind (wind blowing same direction you are flying) and a negative x is a headwind (wind blowing the opposite direction you are flying).

4 Plot your groundspeed (the speed over the ground either adding or subtracting the wind) at various head or tail wind speeds. Use increments of 10.

Insert your coordinate graph below.

D�� I�

1 What does the slope of the line tell you?

2 At what point does your plane fly backwards? Explain.


Chapter 1

Groundspeed is how fast the plane is actually moving if you tracked it with a radar gun from the ground. Airspeed is the speed that the airspeed indicator says on the plane. Airspeed and groundspeed are the same if there is no wind.

Graphs will vary.

The change in the groundspeed.

When -x > y

8 Come Fly with Me!

A�� I�

1 Get a sectional map from your teacher. Using the given scale on the map, determine the distance between two airports.

2 Choose one of your airplanes from the Try It 1 section to fly from one airport to the other using the given sectional map. Using the information for that airplane, find how many hours it will take to travel and how much fuel will be used for the trip.

3 Your teacher will give you a website that pilots use to determine wind speed and direction at various altitudes.

a The first column is a list of airport codes. Find your origin and your destination.

b The columns labeled 3000, 6000, 9000, etc. are the altitudes. You cannot fly over 12500 ft. without oxygen.

c The first two numbers represent the direction the wind is coming from. For this example we are not going to worry about wind direction.

d The next two numbers are the wind speed in knots. Average the wind speed for your origin and your destination airports.

e The final two digits indicate the temperature in Celsius.

4 Using your average wind speed as a head wind, calculate how this affects your time and fuel consumption.

5 Use your average wind speed as a tail wind. Calculate how this affects your time and fuel consumption.

6 What was your reasoning for choosing your airplane? Justify your choice quantitatively.

Chapter 1

http://aviationweather.gov/products/nws/chicago

Answers will vary.

Answers will vary.

Answers will vary.

Answers will vary.

Answers will vary.

Answers will vary.

E�� I�

1 Research one jet that gives the usable fuel in pounds/hr. List the jet you chose and how much usable fuel it can hold.

2 Determine the pounds/hr and convert it to gallons/hr.

3 If you were given the same amount of gallons of gasoline for your automobile, how many miles could you drive on the same amount that was used for the jet?

4 How many times could you travel back and forth between the same airports in your automobile?



Answers will vary.

Answers will vary.

Answers will vary.

Answers will vary.

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